Sources close to the potential bill added that the CBO score was pegged mostly to similar provisions already considered by the nonpartisan panel.

In the Saturday evening meeting among Reid, Boehner, Pelosi and Senate Minority Leader Mitch McConnell (R-Ky.), Reid indicated an openness to considering a two-tranche proposal, but insisted — as Obama has — that the trigger for the second phase not be pegged to another debt limit vote. According to sources, Reid offered two alternative triggers: a bipartisan, bicameral panel would come up with a framework to reduce the deficit further — a scenario similar to a joint commission proposal that Reid has been working on with McConnell — or an idea to instruct Congressional committees to consider the Senate’s bipartisan “gang of six” plan, which gained traction last week.

“I hope that Speaker Boehner and Leader McConnell will reconsider their intransigence,” Reid said in a statement after Saturday’s meeting. “Their unwillingness to compromise is pushing us to the brink of a default on the full faith and credit of the United States. We have run out of time for politics. Now is the time for cooperation.”

A representative for Boehner’s office did not immediately respond Sunday to requests for comment on the sources’ characterization of Reid’s two-track proposals.

Senate sources expressed hope that a dueling-bill situation would not play out this week. But it seems that both Senate and House leaders are inching toward the strategy of passing a bill and daring the other chamber to consider it.

Additionally, aides emphasized that although the $2.7 trillion offer creates another option as time runs out, it “does not foreclose the possibility of a two-step proposal.” They also suggested an openness to amending the proposal, such as adding a deficit reduction panel, if such tweaks could garner GOP votes.

“A commission could be part of it to find deeper cuts and if that’s what it takes to get Republican approval,” a Senate Democratic aide said.

In response to the Democrats’ intention to roll out their own plan, an aide to McConnell said Sunday, “The bipartisan leadership will continue working through the weekend, and we’ll let you know as soon as there’s something to announce.”

Regardless of what leaders do, it’s clear that timing is of the essence. Given the procedural constraints of the Senate and that any objecting Member could stage a filibuster, lawmakers must act within the next few days or risk missing their mark.

“Under the best of circumstances, a controversial bill will take you a week to go through the Senate. And so we need to have an understanding today and move forward,” Senate Majority Whip Dick Durbin (D-Ill.) said Sunday on “Face the Nation.” “I would just say to Speaker Boehner, the president negotiated directly with you in good faith twice, and you walked away from it. At some point, the Speaker has to accept a responsibility beyond his caucus to this nation.”